| L(s) = 1 | − 4·2-s − 2·3-s + 12·4-s + 10·5-s + 8·6-s − 32·8-s − 5·9-s − 40·10-s − 68·11-s − 24·12-s − 52·13-s − 20·15-s + 80·16-s + 164·17-s + 20·18-s − 232·19-s + 120·20-s + 272·22-s − 198·23-s + 64·24-s + 75·25-s + 208·26-s − 26·27-s − 18·29-s + 80·30-s + 196·31-s − 192·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.384·3-s + 3/2·4-s + 0.894·5-s + 0.544·6-s − 1.41·8-s − 0.185·9-s − 1.26·10-s − 1.86·11-s − 0.577·12-s − 1.10·13-s − 0.344·15-s + 5/4·16-s + 2.33·17-s + 0.261·18-s − 2.80·19-s + 1.34·20-s + 2.63·22-s − 1.79·23-s + 0.544·24-s + 3/5·25-s + 1.56·26-s − 0.185·27-s − 0.115·29-s + 0.486·30-s + 1.13·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + p^{2} T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 68 T + 3634 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 p T + 4334 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 164 T + 13606 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 232 T + 26438 T^{2} + 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 198 T + 23785 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 18 T + 33955 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 196 T + 27786 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 160 T + 101082 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 62 T + 72379 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 198 T + 140065 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 164 T + 211426 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 40 T + 5570 p T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 80 T - 85178 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 174 T + 105307 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1054 T + 672761 T^{2} + 1054 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 832 T + 815278 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 820 T + 840150 T^{2} - 820 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 576 T + 880606 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 298 T + 423841 T^{2} + 298 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 182 T + 1152523 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 892 T + 1918278 T^{2} - 892 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.37633716797870581002568657327, −10.00685954993352103908822900158, −9.510833690452697515520259678361, −9.156753639970787623211706907993, −8.310238488988642778454622863177, −8.160438811171503196564085902298, −7.74621079553898103708450547485, −7.39138355439724096841822193351, −6.61956386748368797128677813885, −6.14494642740472718772914267425, −5.69538757554568578164738143350, −5.48748427449309442479610086983, −4.63611812994675585982415437275, −4.02532606305135047173662325231, −2.79462567891229157061265617315, −2.62539810768610520383056663664, −2.01283531906522955885869826120, −1.21712567959737101524827094557, 0, 0,
1.21712567959737101524827094557, 2.01283531906522955885869826120, 2.62539810768610520383056663664, 2.79462567891229157061265617315, 4.02532606305135047173662325231, 4.63611812994675585982415437275, 5.48748427449309442479610086983, 5.69538757554568578164738143350, 6.14494642740472718772914267425, 6.61956386748368797128677813885, 7.39138355439724096841822193351, 7.74621079553898103708450547485, 8.160438811171503196564085902298, 8.310238488988642778454622863177, 9.156753639970787623211706907993, 9.510833690452697515520259678361, 10.00685954993352103908822900158, 10.37633716797870581002568657327
